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Finiteness of perturbative superstring theory

Original title: Patience medal for Jacques Distler

A discussion at shows what a waste of time it is to discuss difficult matters with people who are not capable or honest enough to learn the very basic pre-requisites for such a discussion.

A long "teacup" thread that currently contains 240+ comments has become a place to talk about the finiteness of perturbative string theory. There are many equivalent ways to calculate perturbative superstring amplitudes and many of them have been proven to be finite. Each approach has some advantages and some disadvantages. Different string theorists prefer different approaches but they don't have to agree on this question because formalism is not a part of physics: they only have to agree about the physical predictions such as the cross sections. And you bet they do.

The names associated with the available proofs of the finiteness include Martinec; Mandelstam; Berkovits; Atick, Moore, Sen; d'Hoker, Phong, and others. Some of these papers are more complete - or quite complete - or more constructive than others and there are various causal relations between the papers. Many of these results are secretly equivalent to each other because of the equivalences between the approaches that are demonstrated in other papers. Many of these papers were preceded by less successful papers or papers with flaws - flaws that were eventually fixed and settled.

See also: Four-loop NSR amplitudes
Also, I assure Jacques that he has met people who consider Mandelstam's proof to be a proof, and besides your humble correspondent, this set includes Nathan Berkovits who confirms Mandelstam's proof on page 4 of his own proof in hep-th/0406055, reference 31, even though Nathan's proof is of course better. ;-)

At any rate, the question of perturbative finiteness has been settled for decades. Many people have tried to find some problems with the existing proofs but all of these attempts have failed so far. Nikita will certainly forgive me that I use him as an example that these episodes carry human names: Nikita Nekrasov had some pretty reasonably sounding doubts whether the pure spinor correlators in Berkovits' proof were well-defined until he published a sophisticated paper with Berkovits that answers in the affirmative.

After these long years, the hypotheses that there could be some divergences even in perturbative superstring theory at a finite order have become extraordinary claims that would require extraordinary evidence.

No one has found any repeatable problems with the proofs and most physicists have lost their interest in the finiteness issue because the finiteness has become an obvious fact. Moreover, perturbative string theory has become just a limit of the full string/M-theory that is known to offer a much richer set of physical phenomena. We have a lot of evidence and some proofs that the full string theory is consistent itself - i.e. that string theory is non-perturbatively consistent - and the perturbative consistency has been transformed into a small trivial corollary of the more general fact.

Needless to say, the people who continue to work on these perturbative issues don't like the fact that virtually everyone else views the perturbative finiteness as a closed subject, much like the conventional QCD researchers are not too happy that most high-energy physicists consider QCD to be a well-established theory.

Some of the finiteness experts would like to encourage some doubts about the finiteness among the string theorists which would also encourage the physicists' interest in their additional work on the finiteness. But neither of them has any publishable evidence that the string perturbation theory contains divergent terms. If someone found such evidence, she would immediately publish it because it would be a bombshell discovery - thanks Peter F.! - if it turned out to be correct (which seems extremely unlikely now).

Jacques' favorite proof of the finiteness is the articleby Atick, Moore, and Sen. I will use the acronym AMS even though we have recently argued that AMS cannot mean anything else than the American Mathematical Society: mea culpa. ;-) It's a rather clever and technical argument although it doesn't require any extraordinarily deep ideas. They study the amplitudes with "n" external particles at genus "g" and show that the amplitude is a result of an integral whose integrand is ambiguous because it depends on the gauge choice. However, they can show that the ambiguity is only a total derivative which integrates to zero because it can be expressed as a product of some tadpoles or vacuum energy amplitudes at lower genera "g".

The finiteness of perturbative superstring theory thus follows from the vanishing of the tadpoles and the vacuum graphs. Both of them can be proven by mathematical induction. The only remaining ambiguity may be absorbed into a redefinition of the string tension and/or the string coupling constant. The perturbative string S-matrix is finite and unique up to one field redefinition (for the dilaton) and one scale redefinition, as expected.

In the gauges that AMS considered, the integrals are zero even though the integrands are not. It is therefore an approach that makes certain features of the theory - such as the vanishing of the spacetime cosmological constant - look more complicated than necessary. But if something is more complicated than necessary, it doesn't mean that it can't be proven. And of course that it has been proven in this case.

Phong and d'Hoker have recently found complementary results that are in some sense more beautiful and impressive than the results of AMS. These new results are less general in some respects but more powerful in other respects. In the intersection of their physical results, d'Hoker and Phong fully agree with AMS, of course.

The results of Phong and d'Hoker are not really needed for a proof of finiteness although if you prefer to see explicit and compact formulae for the resulting amplitudes before you agree that the result is finite, i.e. if you hate non-constructive proofs, Phong and d'Hoker is your way to go.

More concretely, the precise method of d'Hoker and Phong only works up to some genus - two (where they have completed the work) and perhaps three - but not beyond. However, this method leads to explicit results for the amplitudes. Moreover, some properties of these amplitudes are more manifest than in the AMS framework. For example, the integrals that define the cosmological constant vanish point by point on the moduli space of supersymmetric Riemann surfaces: not only the integrals are zero but the integrands vanish, too.

But the integrands don't have any invariant physical meaning in spacetime. The only truly physical result are the integrals over the moduli spaces of worldsheets. The integrand is just an intermediate result, and depending on the precise strategy how you calculate and how you fix the gauge symmetry, the integrands may differ. This difference causes no problems.

The relation between d'Hoker and Phong on one side and AMS on the other side is a typical example of the phenomena discussed in Conventions and physics: some approaches are better to see something, other approaches are better to see something else, but if they're really physically equivalent, and the two examples here demonstrably are, we could in principle get the same answers to any physical question using both approaches.

But this trivial fact is another thing that the alternative physicists and critics of science, to use a highly euphemistic label for this kind of people, will never be able or willing to understand. Whenever it's convenient for spreading a particular convenient myth, they are ready to argue that two equivalent approaches are not equivalent, and if two groups of physicists use different gauge choices, they must be in contradiction. That's of course a complete nonsense: in other words, it is a rudimentary misunderstanding of the meaning of the gauge symmetry in physics.

A careful physicist may always have doubts about something. But having private doubts about a proof unsupported by a glimpse of evidence is something very different from the claim that it is likely that there is an error in a paper that has resisted nearly two decades of checks.

Jacques Distler has primarily debated two notorious critics of science who are obsessed with the weird idea - and according to everything we know, false idea - that perturbative string theory may contain divergent or ill-defined terms. Why are they so obsessed with such a highly unlikely speculation? Well, partly it is because both of them have recently published scientifically atrocious books that dedicate dozens of pages to this particular craziness (although the quality of the remainder of these books is comparably low).

One of these people keeps on repeating some rumors that a physicist or mathematician may have privately told him that he may have thought that there could be a problem with the AMS proof, or something like that, except that it is impossible to get even a remote idea what this hypothetical doubt could be based on. This kind of reasoning based on rumors resembles the conspiracy theories about the NASA trip to the Moon being filmed in Hollywood.

The critic had to admit that he had no clue about the content of these physical questions even though on his own weblog, he likes to mislead people into thinking that he has an idea what string theory actually is. This behavior is what I call a morally defective fraud.

These people imagine that the goal of science is to repeat and spread rumors throughout the world, with the help of 38 journalists with high-school physics education and a vitriolic blog posting constant lies, which will eventually lead to the scientific consensus that there could exist inconsistencies in perturbative string theory. Such a politically correct conclusion could be reached while no one actually has to read the papers. ;-) Well, let me disappoint the alternative scientists but that's really not possible.

Even if the critic of science understood correctly that someone could be skeptical about a particular proof, such a piece of information would be largely irrelevant unless this information is supplemented with some technical "meat". What's much more important than a rumor is that at this moment, there exists no known publishable counter-argument that could justify the speculation that there is something wrong with the numerous proofs of finiteness we have. It is easy to see if you check 60 citations of AMS. Rumors are secondary: even if a majority of theorists thought that there could be an error in the proof, the proof is alive until someone actually finds the error. Science can't build on rumors or beliefs much like it can't build on crackpot discussion forums in Manhattan.

Needless to say, this is not how the critics of science understand how science works.

Jacques Distler's understanding of the proofs of the finiteness of string perturbation theory obviously exceeds the knowledge of the critics of science roughly by three orders of magnitude: they probably haven't even seen the paper they try to discuss, and if they did, they have only read it at the level of comparative literature (looking for words such as "disaster", without understanding what these words refer to, that could be useful for their new junk books), not at the level of science.

It is kind of irritating if these debates try to pretend that they are debates among the peers. I would understand if the critics of science sat down and became Jacques' students for a while and if characteristically patient Jacques would try to reduce the gap between his understanding and their understanding from three orders of magnitude to two orders of magnitude, hopefully in the direction up, or something along these lines.

Of course, they would first have to learn which paper by AMS actually contains the proof and maybe even look at it. So far, they haven't been able to do so. If I had a similar talkative undergraduate student in a class of string theory whose approach would be as irrational and dishonest as the approach of the people who try to argue with Jacques, he or she would simply fail my class; they would be explained that high-energy theoretical physics is not compatible with their way of thinking about the world. Unfortunately, this is not what happened with these 2+ people.

Let me finally say that selling such an exchange between Jacques Distler who is undoubtedly among the world's top 50 scholars who have read and understood the stringy perturbative finiteness papers (even though there exist people who have written more important things about the subject) on one side and the critics whose intellectual limitations are breathtakingly low on the other side as a discussion of peers is beyond ridiculous.

And that's the memo.

P.S.: Let me emphasize in advance that I will be erasing nonsensical comments about these integrals and conspiracy theories written by uninformed people and I will leave these vertical discussions painted as horizontal discussions to other blogs - blogs that believe that it is a good idea to be giving legitimacy to the crackpots.

I believe that it would be very dangerous to blur the boundaries between scientists who write and read the papers on one side and crackpots who misinterpret the rumors on the other side. This blog is not another Not Even Wrong and it will never become one.

I have known for many years that certain vicious crackpots will do anything to achieve certain perverse goals, they abuse every single sign of weakness, and it is quite important for the future of science to be very open about their being crackpots instead of trying to be "nice". And I wonder how much more time Clifford et al. will need before they will agree with me.

They might be just beginning to understand that the more support they lend to various conspiracy theories and incorrect statements about theoretical physics, the "nicer" their image painted by the Manhattan supreme blogger will be. Do you really dream about this superficially positive image?

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